Journal Algebra Discrete Math.
cJournal “Algebra and Discrete Mathematics”
Topological semigroups of matrix units
Oleg V. Gutik, Kateryna P. Pavlyk
Communicated by B. V. Novikov
Abstract. We prove that the semigroup of matrix units is
stable. Compact, countably compact and pseudocompact topolo-giesτon the infinite semigroup of matrix unitsBλsuch that(Bλ, τ)
is a semitopological (inverse) semigroup are described. We prove the following properties of an infinite topological semigroup of ma-trix units. On the infinite semigroup of mama-trix units there exists no semigroup pseudocompact topology. Any continuous homomor-phism from the infinite topological semigroup of matrix units into a compact topological semigroup is annihilating. The semigroup of matrix units is algebraically h-closed in the class of topological inverse semigroups. SomeH-closed minimal semigroup topologies on the infinite semigroup of matrix units are considered.
In this paper all topological spaces are Hausdorff.
A semigroup is a set with a binary associative operation. The semi-group operation is called a multiplication. A semigroup S is called in-verse if for any x ∈ S there exists a unique y ∈ S such that xyx = x
andyxy =y. An element y ofS is calledinverseto x and is denoted by
x−1. If S is an inverse semigroup, then the map which takes x ∈ S to the inverse element ofx is called the inversion.
A topological spaceSthat is algebraically semigroup with a separately continuous semigroup operation is called a semitopological semigroup. If the multiplication onSis jointly continuous, thenS is called atopological semigroup.
2000 Mathematics Subject Classification: 20M15, 20M18, 22A15, 54A10, 54C25, 54D25, 54D35, 54H10.
Key words and phrases: semigroup of matrix units, semitopological semigroup, topological semigroup, topological inverse semigroup, H-closed semigroup, absolutely
Journal Algebra Discrete Math.
A topological(semitopological) inverse semigroupis a topological (se-mitopological) semigroup S that is algebraically an inverse semigroup with continuous inversion. Obviously, any topological (inverse) semigroup is a semitopological (inverse) semigroup.If τ is a topology on a (inverse) semigroup S such that (S, τ) is a topological (inverse) semigroup, then τ is called a (inverse) semigroup topology onS.
We follow the terminology of [6, 7, 11, 16], and [22].
IfSis a semigroup, then byE(S)we denote the subset of idempotents of S. By ω we denote the first infinite ordinal. Further, we identify all cardinals with their corresponding initial ordinals.
Let S be a semigroup and Iλ be a set of cardinality λ≥ 2. On the setBλ(S) =Iλ×S1×Iλ∪ {0} we define the semigroup operation′· ′ as follows
(α, a, β)·(γ, b, δ) =
(
(α, ab, δ), if β =γ,
0, if β 6=γ,
and (α, a, β)·0 = 0·(α, a, β) = 0·0 = 0 for α, β, γ, δ ∈ Iλ, a, b ∈ S1. The semigroup Bλ(S) is called a Brandt-Howie semigroup of the weight
λoverS [13] or aBrandtλ-extension of the semigroup S [14]. Obviously
Bλ(S) is the Rees matrix semigroup M0(S1;Iλ, Iλ,M), where Mis the
Iλ×Iλ identity matrix. If a semigroupS is trivial (i.e. ifS contains only one element), then Bλ(S) is the semigroup of Iλ ×Iλ-matrix units [7], which we shall denote by Bλ.
A semigroup B(p, q)generated by elementsp and q which satisfy the condition pq = 1 is called bicyclic. The bicyclic semigroup plays the important role in the Algebraic Theory of Semigroups and in the Theory of Topological Semigroups. For example the well-known O. Andersen’s result [1] states that a (0–) simple semigroup is completely (0–) simple if and only if it does not contain the bicyclic semigroup (see Theorem 2.54 of [7]). L. W. Anderson, R. P. Hunter and R. J. Koch in [2] proved that the bicyclic semigroup cannot be embedded into a stable semigroup. Also any Γ-compact topological semigroup (and hence compact topological semigroup) does not contain the bicyclic semigroup [15] and therefore every (0–) simple Γ-compact topological semigroup is completely (0–) simple.
Journal Algebra Discrete Math.
and this implies the structure theorem for 0-simple compact topological inverse semigroups. Moreover, any continuous homomorphism from the infinite topological semigroup of matrix units into a compact topological semigroup is annihilating. Also we shall prove that if a topological inverse semigroupScontains a semigroup of matrix unitsBλ, thenBλ is a closed subsemigroup ofS, i.e. Bλ is algebraicallyh-closed in the class of topo-logical inverse semigroups. SomeH-closed minimal semigroup topologies on the infinite semigroup of matrix units will be considered.Lemma 1. Let a, b ∈ S1, α, β ∈ I
λ. Then the following conditions are
equivalent:
(i) aS1 ⊆bS1 [S1a⊆S1b];
(ii) (α, a, β)Bλ(S)⊆(α, b, β)Bλ(S) [Bλ(S)(α, a, β)⊆Bλ(S)(α, b, β)].
Proof. (i)⇒(ii).
(α, a, β)Bλ(S) =
= [
γ∈Iλ
(α, aS1, γ)∪ {0} ⊆ [
γ∈Iλ
(α, bS1, γ)∪ {0}= (α, b, β)Bλ(S).
(ii)⇒(i). Let(α, a, β)Bλ(S)⊆(α, b, β)Bλ(S), then
[
γ∈Iλ
(α, aS1, γ)∪ {0} ⊆ [
γ∈Iλ
(α, bS1, γ)∪ {0}
and hence S
γ∈Iλ(α, aS1, γ)⊆ S
γ∈Iλ(α, bS1, γ). Therefore,aS1 ⊆bS1. The proof of equivalency of the dual conditions is similar.
A semigroup S is called stableif and only if
(i) for a, b∈S,Sa⊆SabimpliesSa=Sab;
(ii) for c, d∈S,cS ⊆dcS implies cS=dcS.
Stable semigroups were first investigated by R. J. Koch and A. D. Wal-lace in [17]. A semigroup S is called weakly stable if S1 is stable [20]. Every stable semigroup S is weakly stable [17]. If S is a regular semi-group, then the converse holds. L. O’Carroll [20] proved that the converse does not hold in general.
Theorem 1. A semigroupSis weakly stable if and only ifBλ(S)is stable
Journal Algebra Discrete Math.
Proof. (⇐) SupposeaS1⊆baS1[S1a⊆S1ab] fora, b∈S1. By Lemma 1 we get
(α, a, β)Bλ(S)⊆(α, b, α)(α, a, β)Bλ(S) = (α, ba, β)Bλ(S)
[Bλ(S)(α, a, β)⊆Bλ(S)(α, a, β)(β, b, β) =Bλ(S)(α, ab, β)]
for all α, β ∈Iλ. Since the semigroup Bλ(S) is stable for each cardinal
λ≥2, then
(α, a, β)Bλ(S) = (α, b, α)(α, a, β)Bλ(S) = (α, ba, β)Bλ(S)
[Bλ(S)(α, a, β) =Bλ(S)(α, a, β)(β, b, β) =Bλ(S)(α, ab, β)]
and by Lemma 1, aS1 =baS1 [S1a=S1ab]. Therefore, the semigroup S
is weakly stable.
(⇒) Let λ≥2. Suppose (α, a, β)Bλ(S) ⊆(γ, b, δ)(α, a, β)Bλ(S) for
α, β, γ, δ ∈ Iλ, a, b∈ S1. Obviously, α =γ =δ. Thus (α, a, β)Bλ(S) ⊆
(α, ba, β)Bλ(S). By Lemma 1 we get aS1 ⊆ baS1. Since S is a weakly stable semigroup, then aS1 =baS1. Then by Lemma 1 we get
(α, a, β)Bλ(S) =
= (α, ba, β)Bλ(S) = (α, b, α)(α, a, β)Bλ(S) = (γ, b, δ)(α, a, β)Bλ(S).
The proof of the dual statement is similar.
Therefore, the semigroup Bλ(S) is stable for all cardinals λ≥2.
Corollary 1. For every cardinal λ≥2 the semigroup Bλ is stable.
In [10] C. Eberhart and J. Selden proved that the bicyclic semi-group B(p, q) admits only the discrete Hausdorff semigroup topology. M. O. Bertman and T. T. West generalized this result and showed that any Hausdorff topology τ on B(p, q) such that(B(p, q), τ)is a semitopo-logical semigroup is discrete [5]. Lemma 2 implies that the semigroup of matrix units has similar properties.
Further by 0 we denote the zero of the semigroupBλ.
Lemma 2. Let τ be a topology on Bλ such (Bλ, τ) is a
semitopologi-cal semigroup. Then any nonzero element of Bλ is an isolated point of
(Bλ, τ).
Proof. Since (Bλ, τ) is a semitopological semigroup, every left internal translation ls:Bλ → Bλ and every right internal translation rs:Bλ →
Journal Algebra Discrete Math.
{0} ∪ {(γ, β)|γ ∈Iλ, β∈Iλ\ {α}} are closed in (Bλ, τ), and hence the sets Bλ\l−s1(0) ={(α, γ) |γ ∈Iλ} and Bλ\r−s1(0) = {(γ, α) |γ ∈Iλ} are open in(Bλ, τ)for anyα∈Iλ. Therefore any nonzero element ofBλ is an isolated point of (Bλ, τ).
In [5] M. O. Bertman and T. T. West showed that the bicyclic semi-group is embedded into a compact semitopological semisemi-group. The next example shows that on the infinite semigroup of matrix unitsBλthere ex-ists a topologyτc such that(Bλ, τc) is a compact semitopological inverse semigroup.
Example 1. Letλ≥ω. A topology τc on Bλ is defined as follows:
a) all nonzero elements of Bλ are isolated points in Bλ;
b) B(0) = {A ⊆ Bλ | 0 ∈ Aand |Bλ \A| < ω} is the base of the topology τc at the point 0∈Bλ.
Lemma 3. (Bλ, τc) is a compact semitopological inverse semigroup.
Proof. Obviously,τc is a compact topology on Bλ.
For any U = Bλ\ {(α1, β1), . . . ,(αn, βn)} ∈ B(0), where α1, . . . , αn, β1, . . . , βn∈Iλ we have
1) U1· {(α, β)} = {0}S{(γ, β) |γ ∈ Iλ\ {α1, . . . , αn}} ⊆ U, where
U1 =U\ {(α1, α1), . . . ,(αn, αn)} ∈ B(0);
2) {(α, β)} ·U2 = {0}S{(γ, β) | γ ∈ Iλ \ {β1, . . . , βn}} ⊆ U, where
U2 =U\ {(β1, β1), . . . ,(βn, βn)} ∈ B(0);
3) {(α, β)} · {(γ, δ)}={0} ⊆U if β6=γ;
4) {0} ·U ={0} ⊆U andU · {0}={0} ⊆U;
5) {(α, β)} · {(β, γ)}={(α, γ)};
6) (U3)−1⊆U, whereU3=Bλ\ {(β1, α1), . . . ,(βn, αn)} ∈ B(0).
Therefore,(Bλ, τc)is a compact semitopological inverse semigroup.
Remark 1. In [21] A. B. Paalman-de-Miranda proved that the zero of a compact completely0-simple topological semigroupS is an isolated point inS. Example 1 implies that the zero of a completely0-simple compact semitopological inverse semigroup (Bλ, τ) is not necessarily an isolated point in(Bλ, τ).
Journal Algebra Discrete Math.
Corollary 2. If λ ≥ ω then there exists no other topology τ on Bλdifferent fromτc such that(Bλ, τ)is a compact semitopological semigroup.
A topological space X is called countably compact if any countable open cover ofX contains a finite subcover [11]. A topological space X is calledpseudocompact[discretely pseudocompact] if any locally finite [dis-crete] collection of open subsets of X is finite. Obviously any countably compact space and any discretely pseudocompact space are pseudocom-pact.
Theorem 2. Let λ ≥ ω and let τ be a topology on the semigroup of matrix units Bλ such that (Bλ, τ) is a semitopological semigroup. Then
the following statements are equivalent:
(i) (Bλ, τ) is a compact semitopological semigroup;
(ii) (Bλ, τ) is a countably compact semitopological semigroup;
(iii) (Bλ, τ) is a discretely pseudocompact semitopological semigroup;
(iv) (Bλ, τ) is a pseudocompact semitopological semigroup;
(v) (Bλ, τ) is topologically isomorphic to (Bλ, τc).
Proof. The implications (i) ⇒ (ii), (i) ⇒ (iii), (ii) ⇒ (iv) and (iii) ⇒ (iv)are trivial.
(iv) ⇒ (i) Suppose there exists a topology τ on the infinite semi-group of matrix units Bλ such that (Bλ, τ) is a pseudocompact non-compact semitopological semigroup. Then there exists an open cover
U ={Uα}α∈A which contains no finite subcover. Let Uα1 ∈ U such that Uα1 ∋0. Then the setBλ\Uα1 is infinite. We put
U∗={Uα1} ∪ {x|x∈Bλ\Uα1}.
ThenU∗
is an infinite locally finite family, which contradicts the pseudo-compactness of the topological space (Bλ, τ).
Corollary 2 implies the equivalency (i)⇔(v).
Since the bicyclic semigroup B(p, q) admits only discrete semigroup topology [10], B(p, q) admits no compact (countably compact, pseudo-compact) semigroup topology. The next proposition is a similar result for the infinite semigroup of matrix units and it follows from Theorem 2.
Journal Algebra Discrete Math.
A topological semigroupS is calledΓ-compact ifΓ(x) ={x, x2, x3, . . . , xn, . . .}
is a compact subsemigroup ofSfor everyx∈S. Obviously every compact semigroup is Γ-compact. J. A. Hildebrant and R. J. Koch proved that every Γ-compact topological semigroup and hence compact topological semigroup does not contain the bicyclic semigroup [15]. Since for any element a of the semigroup of matrix units we have either aa = a or
aa= 0, the semigroup of matrix units isΓ-compact.
Question 1. Does there exist a compact topological inverse semigroup which contains the semigroup Bω?
In this paper we give a negative answer to Question 1. Moreover, we show that ifλ≥ω, then every continuous homomorphism of the topolog-ical semigroup Bλ into a compact topological semigroup is annihilating and Bλ as a topological inverse semigroup is absolutely H-closed in the class of topological inverse semigroups.
Lemma 4. Let T be a dense subsemigroup of a topological semigroupS and0 be the zero of T. Then 0 is the zero of S.
Proof. Suppose that there exists a ∈ S \T such that 0 ·a = b 6= 0. Then for every open neighbourhood U(b) 6∋ 0 inS there exists an open neighbourhoodV(a)6∋0inSsuch that0·V(a)⊆U(b). But|V(a)∩T| ≥
ω, and hence0∈0·V(a)⊆U(b), a contradiction with the choice ofU(b). Therefore0·a= 0 for alla∈S\T.
The proof of the equality a·0 = 0 is similar.
Lemma 5. If A is non-singleton in Bλ, then 0∈A·A.
Proof. Suppose |A|= 2. Obviously if0∈A, then0∈A·A.
Let 0 ∈/ A and A = {(α, β),(γ, δ)}. If (α, β), (γ, δ) are idempotents of Bλ, then α = β, γ = δ, β 6= γ, and hence 0 = (α, β) ·(γ, δ). If the set A contains a non-idempotent element (α, β), then α 6= β and
0 = (α, β)·(α, β)∈A·A.
Lemma 6. Letλ≥ωand letBλ be a dense subsemigroup of a topological
semigroup S. Then a·a= 0 for all a∈S\Bλ.
Journal Algebra Discrete Math.
Theorem 3. If λ≥ω, then there exists no semigroup topology τ on Bλsuch that (Bλ, τ) is embedded into a compact topological semigroup.
Proof. Suppose, on the contrary, that there exists a semigroup topology
τ onBλ such that(Bλ, τ)is a subsemigroup of some compact topological semigroup S. By Proposition 1, Bλ is not a closed subsemigroup of S. Without loss of generality we assume that Bλ is a dense subsemigroup of S. We denote X =Bλ\ {0}. By Lemma 2, X is a discrete subspace of Bλ and hence X is a locally compact subspace in Bλ. Thus, by The-orem 3.5.8 [11], J =S\X is a closed subspace of S. Therefore,X is a discrete subspace ofS.
Further, we shall show that J is an ideal of the semigroup S. By Lemmas 5 and 6 it is sufficient to prove that ax, xa, ab ∈ J for every
x∈Bλ\ {0},a, b∈ J \ {0}.
Assume that there exist x ∈ Bλ \ {0} and a ∈ J \ {0} such that
ax = c 6= J. Then c is an isolated point in S. Thus for every open neighbourhoodU(a)6∋0at least one of the following conditions holds
(i) |(U(a)\ {a})·x| ≥ω,
(ii) 0∈(U(a)\ {a})·x.
But c is an isolated point in S, a contradiction. The proof for xa is similar.
Suppose ab=c6∈ J for somea, b∈ J. Thencis an isolated point in
S. For every open neighbourhoodsU(a)6∋0 andU(b)6∋0at least one of the following conditions holds
(iii) |(U(a)\ {a})·(U(b)\ {b})| ≥ω,
(iv) 0∈(U(a)\ {a})·(U(b)\ {b}).
Butc is an isolated point inS, a contradiction. Thus, ab=c∈ J. Therefore, J is a compact ideal of S.
By Theorem A.2.23 [16] the Rees quotient-semigroup S/J is a com-pact topological semigroup. But the semigroupS/J is algebraically iso-morphic toBλ, a contradiction with Proposition 1.
A semigroupSis calledcongruence-free(congruence-simple,h-simple) if it has only two congruences: identical and universal [23]. Such semi-groups E. S. Lyapin [18] and L. M. Gluskin [12] calledsimple. Obviously, a semigroup S is congruence-free if and only if every homomorphism h
of S into an arbitrary semigroup T is an isomorphism "into" or is an annihilating homomorphism (i. e. there exists c∈T such that h(a) =c
for all a∈S).
Journal Algebra Discrete Math.
Corollary 3. The semigroup Bλ is congruence-free for every cardinalλ≥2.
Theorem 3 and Corollary 3 imply
Proposition 2. Let λ ≥ ω. Then every continuous homomorphism of the topological semigroup Bλ into a compact topological semigroup is
an-nihilating.
Recall [8] that a Bohr compactification of a topological semigroup S
is a pair (β, B(S)) such that B(S) is a compact topological semigroup,
β: S → B(S) is a continuous homomorphism, and if g: S → T is a continuous homomorphism ofS into a compact semigroupT, then there exists a unique continuous homomorphism f:B(S) → T such that the diagram
S β //
g
B(S)
f
}
}
zz
zz
zz
zz
T
commutes.
Let S be a topological semigroup. Let {(Tα, ϕα)}α∈A be a family of pairs of compact topological semigroups and continuous homomorphisms
ϕα: S → Tα, respectively, such that ϕα(S) is a dense subsemigroup of
Tα for any α ∈ A. Then B(S) is a subsemigroup of Πα∈ATα (see the proofs of Lemma 2.43 and Theorem 2.44 [6, Vol. 1]), where |A|6 22|S|
. Therefore Proposition 2 implies
Corollary 4. Ifλ≥ω, then the Bohr compactification of the topological semigroup Bλ is a trivial semigroup.
Theorem 4. Let λ be a cardinal ≥ 2 and Bλ be a subsemigroup of a
topological inverse semigroupS. Then Bλ is a closed subsemigroup ofS.
Proof. Ifλ < ω thenBω is finite and henceBλ is a closed subsemigroup ofS.
Suppose λ≥ω.
Let Bλ =S1 in S. Then by Proposition II.2 [10] S1 is a topological
inverse semigroup and by Lemma 4 the zero0of the semigroupBλ is the zero of the semigroup S1.
Let b be any element of S1\Bλ. We consider two cases: b ∈E(S1)
andb∈S1\E(S1).
1) Letb∈E(S1). Then for every open neighbourhoodW(b)6∋0there
Journal Algebra Discrete Math.
W(b). Since |U(b)∩Bλ| ≥ ω, there exist α, β, γ, δ ∈ Iλ such that
(α, β),(γ, δ)∈ U(b), and β 6=γ or α6=δ. Then 0 ∈U(b)·U(b) ⊆
W(b), a contradiction. Therefore,E(S1) =E(Bλ).
2) Let b ∈ S1\E(S1). Then b−1 ∈ S1 \E(S1). Since 0 is the zero
of the topological semigroup S1, then b·b−1 6= 0 and b−1·b 6= 0.
Otherwise, ifb·b−1
= 0 orb−1
·b= 0, thenb=b·b−1
·b= 0·b= 0
or b=b·b−1
·b=b·0 = 0, a contradiction with b∈S1\E(S1).
Therefore, there exist e, f ∈ E(S1) = E(Bλ) such that b·b−1 = e,
b−1·b=f ande6=f. Let W(e)and W(f)be open neighbourhoods of e
andf inS1, respectively, such that06∈W(e)and06∈W(f). Then there
exist disjoint open neighbourhoods U(b) 6∋ 0 and U(b−1
) 6∋ 0 such that
U(b)·U(b−1)⊆W(e)and U(b−1)·U(b)⊆W(f). Since|U(b)∩Bλ| ≥ω and |U(b−1)∩B
λ| ≥ ω, there exist (α, β) ∈ U(b) and (γ, δ) ∈ U(b−1) such that β 6= γ or α 6= δ. Therefore, 0 ∈ U(b)·U(b−1
) ⊆ W(e) or
0∈U(b−1)·U(b)⊆W(f), a contradiction with06∈W(e) and06∈W(f). If e=f the proof of the statement is similar.
The obtained contradictions imply the statement of the theorem.
Since a compact topological semigroup is stable (see Theorem 3.31 [6]) a compact 0-simple topological inverse semigroup S is completely 0 -simple and by Theorem 3.9 [7]Sis algebraically isomorphic to the Brandt
λ-extension of a group. Theorems 3 and 4 imply
Corollary 5. LetS be a compact 0-simple topological inverse semigroup. ThenE(S) is finite.
Definition 1 ([25]). LetS be a class of topological semigroups. A semi-groupS∈ S is calledH-closed in S, ifS is a closed subsemigroup of any topological semigroup T ∈ S which contains S as subsemigroup. If S
coincides with the class of all topological semigroups, then the semigroup
S is called H-closed.
We remark that in [25] H-closed semigroups are called maximal. Theorem 4 implies
Corollary 6. Letλbe a cardinal≥2andτ be a semigroup inverse topol-ogy on Bλ. Then (Bλ, τ) is H-closed in the class of topological inverse
semigroups.
Definition 2 ([26]). LetS be a class of topological semigroups. A topo-logical semigroupS∈ S is calledabsolutelyH-closed in the classS if any continuous homomorphic image of S into T ∈ S is H-closed in S. If S
coincides with the class of all topological semigroups, then the semigroup
Journal Algebra Discrete Math.
Corollary 3 and Theorem 4 implyCorollary 7. Letλbe a cardinal≥2andτ be a semigroup inverse topol-ogy onBλ. Then(Bλ, τ)is absolutely H-closed in the class of topological
inverse semigroups.
Let S be a class of topological semigroups. A semigroup S is called
algebraically h-closed in S if S with discrete topology dis absolutely H -closed inS and(S, d)∈ S. IfS coincides with the class of all topological semigroups, then the semigroup S is calledalgebraically h-closed.
AbsolutelyH-closed semigroups and algebraicallyh-closed semigroups were introduced by J. W. Stepp in [26]. There they were calledabsolutely maximaland algebraic maximal, respectively.
Corollary 7 implies
Proposition 3. For any cardinalλ≥2the semigroupBλ is algebraically
h-closed in the class of topological inverse semigroups.
The following example shows that Bω with the discrete topology is notH-closed.
Example 2. LetBω =Iω×Iω∪ {0} be the semigroup of matrix units anda6∈Bω. Let S=Bω∪ {a}. We put
a·a=a·0 = 0·a=a·(α, β) = (α, β)·a= 0
for all(α, β)∈Bω\ {0}.
Further we enumerate the elements of the setIω by natural numbers. Let An = {(2k−1,2k) | k ≥n} for each n ∈ N. A topology τ on S is defined as follows:
1) all points of Bω are isolated in S;
2) B(a) ={Un(a) ={a} ∪An |n∈ N} is the base of the topology τ at the pointa∈S.
Then
a) {(l, m)} ·Un(a) =Un(a)· {(l, m)} ={0} for all(l, m) ∈ Bω\ {0},
n≥max{l, m};
b) Un(a)·Un(a) =Un(a)· {0}={0} ·Un(a) ={0} for anyn∈N;
c) Un(a) is a compact subset of S for eachn∈N.
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Remark 2. Let λ1 and λ2 be cardinals and λ1 ≤ λ2. Then Bλ1 is asubsemigroup of Bλ2. Example 2 implies that for any infinite cardinalλ
the semigroup Bλ is notH-closed.
Definition 3. A Hausdorff topological (inverse) semigroup(S, τ)is said to beminimalif no Hausdorff semigroup (inverse) topology onSis strictly contained inτ. If(S, τ) is minimal topological (inverse) semigroup, then
τ is called minimal semigroup (inverse) topology.
The concept of minimal topological groups was introduced indepen-dently in the early 1970’s by Do¨ıtchinov [9] and Stephenson [24]. Both au-thors were motivated by the theory of minimal topological spaces, which was well understood at that time (cf. [4]). More than 20 years earlier L. Nachbin [19] had studied minimality in the context of division rings, and B. Banaschewski [3] investigated minimality in the more general set-ting of topological algebras.
Question 2 (T. O. Banakh). Is it true that for any cardinal λ≥ω the semigroup Bλ admits minimal (inverse) semigroup topology?
For each α, β ∈Iλ we define
Vα=Bλ\ {(α, γ)|γ ∈Iλ} and Hβ =Bλ\ {(γ, β)|γ ∈Iλ}.
Put
Uα1,...,αn = n \
i=1
Vαi, Uβ1,...,βm = m \
i=1
Hβi and
Uα1,...,αn β1,...,βm =U
α1,...,αn∩U
β1,...,βm,
where α1, . . . , αn, β1, . . . , βm ∈ Iλ, n, m ∈ N. Further we define the following families
Bmv ={Uα1,...,αn|α1, . . . , αn∈Iλ, n∈N} ∪ {(α, β)|α, β∈Iλ},
Bmh={Uβ1,...,βm |β1, . . . , βm∈Iλ, m∈N} ∪ {(α, β)|α, β ∈Iλ},
Bmi={Uβα11,...,β,...,αmn |α1, . . . , αn, β1, . . . , βm∈Iλ, n, m∈N}∪
∪{(α, β)|α, β∈Iλ}.
Obviously, the conditions (BP1)—(BP3) [11] hold for families Bmv,
BmhandBmi, and henceBmv,BmhandBmiare bases onBλ of topologies
τmv,τmh andτmi, respectively.
Lemma 7. Let λ be an infinite cardinal. Then
Journal Algebra Discrete Math.
(ii) (Bλ, τmh) is a topological semigroup;
(iii) (Bλ, τmi) is a topological inverse semigroup.
Proof. (i) Since the following conditions hold
Uα1,...,αn·Uα1,...,αn ⊆Uα1,...,αn, (α, β)·Uα1,...,αn,β ={0} ⊆Uα1,...,αn,
Uα1,...,αn
·(α, β) ={0} ∪ {(γ, δ)|γ ∈Iλ\ {α1, . . . , αn}} ⊆Uα1,...,αn
for every open neighbourhood Uα1,...,αn of the zero of B
λ and for any
(α, β)∈Bλ\ {0},(Bλ, τmv) is a topological semigroup.
The proof of statement (ii) is similar to the proof of item (i). (iii) For every open neighbourhoodUα1,...,αn
β1,...,βm of the zero ofBλ and for any (β, α)∈Bλ\ {0}we have:
Uα1,...,αn β1,...,βm·U
α1,...,αn β1,...,βm ⊆U
α1,...,αn
β1,...,βm, (β, α)·U α1,...,αn
α,β1,...,βm ={0} ⊆U α1,...,αn β1,...,βm,
Uβ,α1,...,αn
β1,...,βm ·(β, α) ={0} ⊆U α1,...,αn β1,...,βm,
Uβ1,...,βm α1,...,αn
−1
⊆Uα1,...,αn β1,...,βm.
Therefore,(Bλ, τmi)is a topological inverse semigroup.
We remark that τmv,τmh andτmiare not locally compact topologies onBλ for λ≥ω.
For A ⊆ Iλ and a ∈Iλ we denote Aα ={(α, β) ∈Bλ |β ∈A} and
Aα ={(β, α)∈Bλ |β ∈A}.
Lemma 8. Let λbe an infinite cardinal, Bλ be a topological semigroup,
andAα [A
α] be a closed subset in Bλ for some α∈Iλ. Then Aβ [Aβ] is
a closed subset ofBλ for any β ∈Iλ.
Proof. Since Bλ is a topological semigroup, then the map λ(α,β):Bλ →
Bλ [ρ(β,α): Bλ → Bλ] defined by the formula λ(α,β)(x) = (α, β) ·x
[ρ(β,α)(x) = x ·(β, α)] is continuous. Therefore Aβ = λ(α,β)
−1
(Aα) [Aβ = ρ(β,α)
−1
(Aα)] is a closed subset ofBλ.
For any A⊆Iλ,α1, . . . , αn, β1, . . . , βm∈Iλ,n, m∈N we denote
Uα1,...,αn(A) =Uα1,...,αn∪ {(α
i, x)|x∈A, i= 1, . . . , n},
Uβ1,...,βm(A) =Uβ1,...,βm∪ {(x, βi)|x∈A, i= 1, . . . , m},
U(α1, . . . , αn;A) =Uα1,...,αn(A)∩Uα1,...,αn(A).
Journal Algebra Discrete Math.
Theorem 5. Letλbe an infinite cardinal. Then the following conclusions hold:(i) τmv is a minimal semigroup topology onBλ;
(ii) τmh is a minimal semigroup topology on Bλ;
(iii) τmi is the coarsest semigroup inverse topology on Bλ, and hence is
minimal semigroup inverse.
Proof. (i) Suppose that there exists a Hausdorff semigroup topology τ0
on Bλ which is coarser than τmv. LetV0 be an element of a base of the
topology τ0 at the zero of Bλ. Then by Lemma 8 there exist an infinite
subsetAinIλ andα1, . . . , αm∈Iλ(m∈N) such thatUα1,...,αm(A)⊆V0.
For any β ∈Iλ,γ ∈Iλ\ {α1, . . . , αm} the following conditions hold:
a) (αi, β) = (αi, δ)·(δ, β), whereδ ∈A\ {α1, . . . , αm}and, obviously,
(αi, δ),(δ, β)∈Uα1,...,αm(A) (i= 1, . . . , m);
b) (γ, β) = (γ, γ)·(γ, β), and (γ, γ),(γ, β)∈Uα1,...,αm(A).
Thus,
Bλ=Uα1,...,αm(A)·Uα1,...,αm(A)⊆V0·V0
for any element V0 of a base of the topology τ0 at the zero of Bλ. This
gives a contradiction with the continuity of the semigroup operation in
(Bλ, τ0). Therefore(Bλ, τmv) is a minimal topological semigroup. The proof of item (ii) is similar to the proof of item (i).
(iii) Let τ be any Hausdorff semigroup inverse topology on Bλ. We define the maps: ϕ: Bλ → E(Bλ) and ψ: Bλ → E(Bλ) by formulae
ϕ(x) = xx−1 and ψ(x) =x−1x. Since the topology τ is Hausdorff then
the sets ϕ−1
((α, α)) = {(α, γ) | γ ∈ Iλ} and ψ−1((α, α)) = {(γ, α) |
γ ∈ Iλ} are closed for each α ∈ Iλ, and hence Uβα11,...,β,...,αmn ∈ τ for all α1, . . . , αn, β1, . . . , βm ∈Iλ. Therefore τmi⊆τ.
Theorem 6. Let λ be an infinite cardinal. Then (Bλ, τmv), (Bλ, τmh),
(Bλ, τmi) areH-closed topological semigroups.
Proof. We shall show that the semigroup (Bλ, τmi) is H-closed. The proofs of H-closedness of the semigroups (Bλ, τmh) and (Bλ, τmv) are similar.
Suppose that there exists a topological semigroup S which contains
(Bλ, τmi)as a non-closed subsemigroup. Then there existsx∈Bλ\Bλ ⊆
S. By Lemma 4, x·0 = 0·x = 0. Then for every open neighbourhood
W(0) inS there exist open neighbourhoods U(0), V(0), and V(x) in S
Journal Algebra Discrete Math.
W(0),V(x)·V(0)⊆U(0), andV(0)·V(x)⊆U(0). We can suppose that
Uα1,...,αn
β1,...,βm =U(0)∩Bλ for some α1, . . . , αn, β1, . . . , βm∈Iλ. Since |V(x)∩Bλ| ≥ω, one of the following conditions holds:
1) the set Bi0 = V(x)∩ {(αi0, γ) | γ ∈ Iλ} is infinite for some i0 ∈
{1, . . . , n};
2) the set Bj0 = V(x)∩ {(γ, α
j0) | γ ∈ Iλ} is infinite for some j0 ∈
{1, . . . , m}.
In the first case we put
Γi0 ={γ ∈Iλ|(αi0, γ)∈V(x)}.
Then the set {(γ, γ) |γ ∈Γi0} ∩U
γ1,...,γk
δ1,...,δl is infinite for any basic neigh-bourhoodUγ1,...,γk
δ1,...,δl ,γ1, . . . , γk, δ1, . . . , δl ∈Iλ. Thus
Bi0 ·U
γ1,...,γk δ1,...,δl 6⊆U
α1,...,αn β1,...,βm,
a contradiction withV(x)·V(0)⊆U(0). In the other case we put
Γj0
={γ ∈Iλ|(γ, αj0)∈V(x)}.
Then the set {(γ, γ) | γ ∈ Γj0} ∩ Uγ1,...,γk
δ1,...,δl is infinite for every basic neighbourhoodUγ1,...,γk
δ1,...,δl ,γ1, . . . , γk, δ1, . . . , δl∈Iλ. Hence
Uγ1,...,γk δ1,...,δl ·B
j0
6⊆Uα1,...,αn β1,...,βm,
a contradiction withV(0)·V(x)⊆U(0).
Therefore the topological semigroup(Bλ, τmi) isH-closed.
Theorem 6 and Corollary 3 imply
Corollary 8. Letλbe an infinite cardinal. Then(Bλ, τmv)and(Bλ, τmh)
are absolutely H-closed topological semigroups.
Theorem 7. For every cardinal λ ≥ω any continuous homomorphism from(Bλ, τmv) [(Bλ, τmh)]into a locally compact topological semigroupS
is annihilating.
Journal Algebra Discrete Math.
By Theorem 6, h(Bλ) is a closed subsemigroup of S and by Theo-rem 3.3.8 [11],h(Bλ)is a locally compact topological semigroup. This is a contradiction with the fact thatτmv is not a locally compact semigroup topology on Bλ.The proof of the theorem for the semigroup (Bλ, τmh)is similar.
Acknowledgement. The authors are deeply grateful to T. Banakh and I. Guran for useful discussions of the results of this paper.
Add to Proof. The authors express their sincere thanks to the referee for very careful reading the manuscript and valuable remarks im-proving the presentation.
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Contact information
O. V. Gutik Department of Algebra, Pidstryhach Insti-tute for Applied Problems of Mechanics and Mathematics of National Academy of Sci-ences, Naukova 3b, Lviv, 79060, Ukraine, and Department of Mathematics, Ivan Franko Lviv National University, Univer-sytetska 1, Lviv, 79000, Ukraine
E-Mail: ovgutik@yahoo.com, topology@franko.lviv.ua
K. P. Pavlyk Department of Algebra, Pidstrygach Insti-tute for Applied Problems of Mechanics and Mathematics of National Academy of Sci-ences, Naukova 3b, Lviv, 79060, Ukraine
E-Mail: katyapavlyk@rambler.ru
